SWTJC STEM – ENGR 1201

Content Goal 13

Quality Factors

1. Accuracy refers to how close the reported value comes to the true value.

2. Precision refers to the clustering of a group of measurements.

Good modeling requires good data. To achieve this, the physical properties of a model must be measured and reported with quality. How do scientists and engineers express measurement quality?

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Content Goal 13

Accuracy

The accuracy of a measurement refers "to how close the reported value comes to the true value". 

Two common methods for expressing accuracy are

I. Mathematical error, also called percent error.

II. Range of uncertainty, also called tolerance. 

Mathematical error is reported in three different ways:

Given:

reported value - the measured value expected value - the theoretically correct or desired value

Then:

a. absolute error = reported value - expected value

b. relative error = absolute error / expected value

c. percent error = relative error · 100%

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Content Goal 13

Mathematical Error

Given:reported value = 122 mmexpected value = 120 mm

Then:a. absolute error = reported value - expected value absolute error = 122 mm – 120 mm = 2 mm Ans

b. relative error = absolute error / expected value relative error = 2 mm / 120 mm =  0.017 Ans Note: relative error has no units.

c. percent error = relative error · 100% percent error = 0.017 · 100% =  1.7 % Ans

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Mathematical Error

The following are general classifications for errors:

1. For consumer purposes, 5-10% error is acceptable

2. For engineering purposes, 1% error is acceptable

3. For scientific purposes, 0.1% error is acceptable

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Content Goal 13

Classification of Error

Range of uncertainty is reported as a nominal value plus or minus an amount called the tolerance.

Reported value: 120 mm ±1 mm = 119 mm to 121 mm

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Range of Uncertainty

nominal value tolerance range of uncertainty

Range of uncertainty is also reported as a nominal value plus or min Range of uncertainty is reported as a nominal value plus or minus an amount called the tolerance us an percent tolerance. Reported value 120 mm ±2% = 117.6 mm to 122.4 mm

Note: 2% of 120 = 2.4, 120 - 2.4 = 117.6, 120 +2.4 = 122.4

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Content Goal 13

Range of Uncertainty

nominal value tolerance range of uncertainty

Actually, there is a third, less formal way of reporting accuracy. The digit representing the smallest value reported implies that the measurement is “accurate to that digit”. If it were not, then why report it? A reported value 14.02 m implies the measurement was accurate to the nearest 0.01m (a cm).

A reported value 14.00 m also implies the measurement was accurate to the nearest 0.01m (a cm).

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Informal Accuracy Reporting

The precision of a measurement refers to

" the clustering of a group of measurements and can be determined by calculating standard deviation, standard error, or confidence interval". 

Precision is usually expressed by the number of significant digits reported for the magnitude of the measurement. 

3 significant digits is good; 5 significant digits is excellent. 

A weighing scale that reports 128 grams is less precise than one that reports 128.35 grams. 

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Content Goal 13

Precision

Precision in measurement comes at a price!

3 significant digits, commercial quality – Costs $100

4 significant digits, industrial quality – Costs $500

5 significant digits, scientific quality  - Costs $2500

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Precision Costs $’s

Measurement precision must be interpreted in light of measurement accuracy.  Let’s use a target practice example:

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Content Goal 13

Precision – Target 1

The best situation, the shots are tightly clustered (high precision) on the center circle (high accuracy).

Measurement precision must be interpreted in light of measurement accuracy.  Let’s use a target practice example:

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Precision – Target 2

The next situation, shots are near the center (high accuracy), but not tightly clustered (low precision).  

Measurement precision must be interpreted in light of measurement accuracy.  Let’s use a target practice example:

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Precision – Target 3

In the next situation, a tight cluster (high precision) is far off center (low accuracy).

Measurement precision must be interpreted in light of measurement accuracy.  Let’s use a target practice example:

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Precision – Target 4

Finally, widely scattered shots (low precision) appear away from the center (low accuracy).

Which is the best and which is worst?

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Precision - Comparison

Best Most InsidiousWhy?

Worst

Errors in measurement can be classified in two types:

1. Systematic Errors

“result from an inherently wrong measurement”

2. Random Errors

“result from externally introduced factors”

SWTJC STEM – ENGR 1201

Content Goal 13

Error Types

Systematic errors

“result from an inherently wrong measurement”

The measuring instrument and/or procedure is flawed in some specific way.  One characteristic of systematic errors is that they are generally either positive or negative and of roughly constant value.

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Systematic Errors Tires

Example I

Jake has decided to use oversize tires on his truck, but his dad reminds him this could affect the speedometer and odometer readings.  Explain.

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Systematic ErrorsTires

Speedometer and odometer operation is based on number of tire rotations. The new tire will make fewer revolutions. Why?

Old Tire New Tire

Show that a 10% increase in tire diameter will cause a 10% low reading in speed and distance. (Exam question)

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Systematic ErrorsBarometric Pressure

Example II

San Juana has been assigned a school project to record atmospheric pressure at noon for a two week period and plot the data with Excel.  She learns of an Internet site operated by her school that provides hourly data including atmospheric pressure.  So for two weeks she links to the site and records the data in a spreadsheet.  She creates the chart and prepares to turn in the project when she learns that the school’s electronic barometer was improperly calibrated and was reading 0.05 inches of Hg too high.  Why is this a systematic error?

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Systematic ErrorsBarometric Pressure

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Random Errors

Random errors are “not inherent to the measuring process”. 

Frequently they are introduced by external factors that cause a scattering of the measured data.

When the scattering is distributed equally about the true value, the error can be mitigated somewhat by making multiple measurements and averaging the data. 

• Vibration in mechanical devices produces random errors.

•  In electronic devices, noise produces random errors.   

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Random ErrorsWave Action

Example I

Jane is working as an intern for the national park service during the summer and has been assigned the task of monitoring the water level at Lake Amistad near Del Rio, Texas.  A measurement rod at the end of a pier is marked with depth graduations to make the job easier.  At 11:00 AM the first day she prepares to take the daily measurement, but immediately encounters a problem.  Wind induced wave action on the lake causes the water level to rise and fall on the measurement rod making it difficult to read. Why is the error random? How can she mitigate it.

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Random ErrorsWave Action

Measuring Rod

Mean Wave Height (M)

Low Water Point (L)

High Water Mark (H)

Typical Wave

Jane Reports: M = (H + L) /2

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Random ErrorsEngine Test

Example II

Richard is helping test engine performance at Southwest Research Labs in San Antonio.  Fuel pressure is being monitored electronically and recorded using a computer.  The data is copied to Excel and plotted.  A typical recorded segment is shown in the chart below.  Because of engine vibration effects on the pressure sensor as well as electronically induced noise, the fuel pressure data is somewhat erratic (blue line).  The effects are clearly random in nature so he proposes to add an averaging circuit to "smooth" the output.  After doing so, the fuel pressure data plot is much smoother (red line).  By averaging out the error, the resulting data is more representative of the actual fuel pressure.

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Random ErrorsEngine Test

SWTJC STEM – ENGR 1201

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Quality Factors

1. Accuracy "refers to how close the reported value comes to the true value“

2. Precision "refers to the clustering of a group of measurements“

Scientists and engineers express measurement quality using,

Errors in measurement can be classified in two types:

1. Systematic Errors results from a measurement that is inherently wrong.

2. Random Errors results from the effect of external factors.

SWTJC STEM – ENGR 1201

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Error Types